Most of the charts on this site have a logarithmic vertical axis. That is not a stylistic preference. It is the only axis on which the thing we are actually arguing about — the rate of change — is visible at all.

This piece is the onramp. If you already read slope on a log axis as a growth rate, you can skip it; everything else here assumes you do.

Ordinary graph paper is a grid, everywhere

The Cartesian plane we all learned to draw on is linear: the same distance means the same amount everywhere on the page. An inch is an inch at the bottom of the axis and an inch at the top.

That sounds like the only sensible way to build an axis, and for most things it is. Lay a ten-foot square on the ground in Rochester and lay another one in Helsinki, and they are the same square. Our physical intuition is built on a uniform grid, and linear axes match it.

The trouble starts when the quantity you are plotting does not grow by adding a fixed amount but by multiplying by a fixed factor. On a uniform grid, multiplication does not look like anything until it suddenly looks like everything.

The everyday exponential: compound interest

Compound interest is the exponential that most people have already met. Put $1,000 in an account, leave it, and each year’s interest is computed on a balance that already includes every previous year’s interest. The growth compounds because last year’s gain is part of this year’s base.

Here are three rates (3%, 5%, 7%) on $1,000, over twenty-five years, on a normal linear axis.

Two things fall out of that picture, and they are the reason compound interest gets taught at all.

Time is the dominant term. After one year, 7% beats 3% by forty dollars. After forty years it beats it by a factor of 4.6: $14,974 against $3,262. Money invested early does work that money invested late cannot catch up to without a much larger deposit.

The rate is the other term, and it is not a small lever. A difference of four percentage points does not produce a 4% better outcome. Over a long enough run it produces a different order of magnitude. The whole argument of this site is a claim about a rate, which is why we spend so much effort making rates legible.

Stretch the same chart and the axis breaks

Nothing about those three curves changes if we let them run longer. But the chart stops working almost immediately.

This is the failure mode, and it is worth naming precisely: a linear axis has to be scaled to the largest value on it, and an exponential’s largest value is enormous relative to everything else it ever was. So the tall end of the curve consumes the whole axis and every earlier decade — every part of the trend where the interesting comparisons live — gets pressed into the bottom pixel row.

A reader looking at that chart would conclude that the 5% account is worthless. It is worth $1.5 million.

What a log axis does

A logarithmic axis changes the rule for what distance means. Instead of equal distance = equal amount added, it is equal distance = equal factor multiplied. Each major gridline is ten times the one below it, not ten units more.

Same data, same three rates, same 150 years:

That is the whole trick, and it produces four reading rules worth internalizing:

• A straight line means a constant growth rate. Not constant growth — constant rate. The dollars added per year are rising the entire time.
• The slope is the rate. Steeper is faster. You can read a growth rate off a log chart with a ruler.
• Parallel lines are growing at the same rate, whatever their absolute levels.
• A bend is news. If a line that was straight starts curving, the rate itself changed, and that is usually the only event on the chart worth arguing about.

The last one is the reason we use this axis. On a linear plot every exponential bends upward, so a bend tells you nothing. On a log plot, bending is rare and it means something specific.

Next time a stock chart in Yahoo Finance offers you a linear/log toggle, flip it. The log view is the one where a decade of steady 8% and a flat decade followed by a spike look different from each other.

The same principle on real data: transistor counts

Compound interest is a clean case because the rate is defined in advance. Real data is messier, and this is where the log axis earns its keep. The straightness of the line is now a finding rather than an assumption.

Here are nine flagship processors spanning fifty-one years, plotted the same way twice.

Two charts, one dataset, opposite conclusions. From the linear panel you would conclude that semiconductor progress is a recent event and that everything before it was a flat prelude. From the log panel you would conclude that the industry has held a near-constant exponential for fifty years, which is the actual claim Moore’s Law makes, and the planning assumption an entire industry has run on.

The linear panel is not lying. It is answering a different question: how big is the difference in absolute terms? That question has a legitimate answer, and it is “unimaginably big.” But if you want to know whether the rate changed, the linear panel cannot tell you, and the log panel can.

What the log axis costs you

It is not a free upgrade, and knowing the price is part of reading one properly.

• It compresses the drama. A gap of one gridline looks modest and is a factor of ten. Readers who skim a log chart routinely under-feel the magnitudes. That is the exact inverse of the linear chart’s failure, and it is why the captions on this site usually name the endpoint values in words.
• Zero does not exist on it, and neither do negative numbers. A log axis can approach zero forever without reaching it, so anything that crosses zero needs a different treatment.
• It flatters a bad fit. Almost anything looks roughly linear once you compress it across six decades. Straightness on a log axis is evidence of a constant rate, but “roughly straight over five orders of magnitude” can still hide a factor-of-two miss.

Back to the charts this site runs on

The GDP-per-capita chart in the System of the World is exactly the chart type built above. Its vertical axis doubles at each gridline, its three fitted segments are straight lines, and the claim being made is entirely about their slopes: roughly flat for eighteen centuries, about 0.9% a year through industrialization, about 2.1% a year since 1950.

On a linear axis that chart would be a single vertical spike at the right-hand edge and eighteen centuries of flat baseline: visually dramatic, and useless for the thing under discussion, which is whether the modern rate is a new regime and whether it is holding.

The bend is the news. That is what the axis is for.